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The least slope of a convex function and the maximal monotonicity of its subdifferential


In this paper, we give a direct proof of Rockafellar's result that the subdifferential of a proper convex lower-semicontinuous function on a Banach space is maximal monotone. Our proof is simpler than those that have appeared to date. In fact, we show that Rockafellar's result can be embedded in a more general situation in which we can quantify the degree of failure of monotonicity in terms of a quotient like the one that appears in the definition of Fréchet differentiability. Our analysis depends on the concepts of the least slope of a convex function, which is related to the steepest descent of optimization theory.

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The author would like to express his thanks to R. R. Phelps for reading a preliminary version of this paper and making some very valuable suggestions.

Communicated by O. L. Mangasarian

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Simons, S. The least slope of a convex function and the maximal monotonicity of its subdifferential. J Optim Theory Appl 71, 127–136 (1991).

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Key Words

  • Subdifferential of a convex function
  • maximal monotone operator
  • separation theorem
  • sandwich theorem